%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Miscellaneous%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\marg}[2]{\addtolength{\marginparwidth}{0.2in}
%                  \marginpar{\textcolor{red}{{#1}\textit{$\leftarrow$#2}}}}
%\newcommand{\loccit}{\emph{loc.cit.}}

%\newcommand{\fn}{f_{\text{norm}}}

\newcommand{\emphr}[1]{\emph{\red #1}}


\newcommand{\rec}[1]{\frac{1}{#1}}
\newcommand{\eqdef}{\stackrel{\Delta}{=}}
\newcommand{\dtheta}{\frac{\partial }{\partial \theta}}
\newcommand{\Etheta}{E_{\theta}}
\newcommand{\etal}{\emph{et.\,al.}\xspace}
\newcommand{\rect}{\operatorname{rect}}
\newcommand{\corr}{\operatorname{corr}}
\newcommand{\Cov}{\operatorname{Cov}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\pons}{\operatorname{PONS}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\mpons}{\operatorname{\mu PONS}}
\newcommand{\intR}{\ensuremath{\int_{\bR}}}
\newcommand{\cbr}[1]{\cB^{\geq #1}}
\newcommand{\id}{\operatorname{id}}
%\newcommand{\ave}{\operatorname{ave}}
\newcommand{\chr}{\operatorname{Char}}
\newcommand{\prim}{\operatorname{Prim}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\nilp}{\operatorname{Nilp}}
\newcommand{\aug}{\operatorname{Aug}}
\newcommand{\erf}{\text{erf}\;}
\newcommand{\mse}{\text{mse}\;}
\newcommand{\ltwor}{\ensuremath{L^2(\bR)}}
\newcommand{\ltwo}[1]{\ensuremath{L^2(#1)}}
\newcommand{\ltwortwo}{\ensuremath{L^2(\bR^2)}}
\newcommand{\var}{\text{var}}
\newcommand{\ave}{\operatorname{ave}}
\newcommand{\sgn}{\text{sgn}\;}
\newcommand{\tr}{\text{tr}\;}
\newcommand{\diag}{\text{diag}\;}
\newcommand{\mbif}[1]{\text{ if $#1$}}
\newcommand{\spn}{\operatornamewithlimits{span}}
\newcommand\ket[1]{\protect{\mid\!#1\rangle}}
\newcommand\bra[1]{\protect{\langle#1\!\mid}}
\newcommand\innerprod[2]{\protect{\langle {#1}\!\mid\! {#2}\rangle}}
\newcommand\average[3]{\bra{#2}#1\ket{#3}}
\newcommand{\ppartial}[1]{\frac{\partial}{\partial #1}}
\newcommand\expect[1]{\langle #1\rangle}
\newcommand\ketnil{\ket{0}}
\newcommand\braone{\bra{1}}
\newcommand\branil{\bra{0}}
\newcommand\ketone{\ket{1}}
\newcommand\half{\ensuremath\frac{1}{2}}
\newcommand\recsqrt[1]{\ensuremath{\frac{1}{\sqrt{#1}}}}
\newcommand\spinup{\ensuremath\ket{{\,\uparrow}}}
\newcommand\spindown{\ensuremath\ket{{\,\downarrow}}}
\newcommand\spnup[1]{\ensuremath\ket{{\,\uparrow_{#1}}}}
\newcommand\spndown[1]{\ensuremath\ket{{\,\downarrow_{#1}}}}
\newcommand\bspnup[1]{\ensuremath\bra{{\uparrow_{#1}}\,}}
\newcommand\bspndown[1]{\ensuremath\bra{{\downarrow_{#1}\,}}}
\newcommand{\adj}[1]{\ensuremath {#1}^{\dag}}
\newcommand{\SUtwo}{\ensuremath{\text{SU}(2)}}
\newcommand{\SOthree}{\ensuremath{\text{SO}(3)}}
\newcommand{\sutwo}{\ensuremath{\frak{su}(2)}}
\newcommand{\argmax}{\operatornamewithlimits{argmax}}
\newcommand{\argmin}{\operatornamewithlimits{argmin}}
\newcommand{\loccit}{\emph{loc.cit.}\xspace}

%%%%%%%%%%%%%%%%%Colors%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\providecommand{\red}{\color{red}}
\providecommand{\blue}{\color{blue}}
\providecommand{\green}{\color{green}}
\providecommand{\black}{\color{black}}
\providecommand{\white}{\color{white}}
\providecommand{\yellow}{\color{yellow}}

%%%%%%%%%%%%%%%%%%%%%Fraktur%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\fH}{\ensuremath{{\mathfrak H}}}
\newcommand{\fK}{\ensuremath{{\mathfrak K}}}
\newcommand{\llie}[1]{\ensuremath{{\mathfrak #1}}}


%%%%%%%%%%%Bold roman caps%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\bA}{\ensuremath{{\mathbf A}}}
\newcommand{\bB}{\ensuremath{{\mathbf B}}}
\newcommand{\bC}{\ensuremath{{\mathbf C}}}
\newcommand{\bD}{\ensuremath{{\mathbf D}}}
\newcommand{\bE}{\ensuremath{{\mathbf E}}}
\newcommand{\bF}{\ensuremath{{\mathbf F}}}
\newcommand{\bG}{\ensuremath{{\mathbf G}}}
\newcommand{\bH}{\ensuremath{{\mathbf H}}}
\newcommand{\bI}{\ensuremath{{\mathbf I}}}
\newcommand{\bJ}{\ensuremath{{\mathbf J}}}
\newcommand{\bK}{\ensuremath{{\mathbf K}}}
\newcommand{\bL}{\ensuremath{{\mathbf L}}}
\newcommand{\bM}{\ensuremath{{\mathbf M}}}
\newcommand{\bN}{\ensuremath{{\mathbf N}}}
\newcommand{\bO}{\ensuremath{{\mathbf O}}}
\newcommand{\bP}{\ensuremath{{\mathbf P}}}
\newcommand{\bQ}{\ensuremath{{\mathbf Q}}}
\newcommand{\bR}{\ensuremath{{\mathbf R}}}
\newcommand{\bS}{\ensuremath{{\mathbf S}}}
\newcommand{\bT}{\ensuremath{{\mathbf T}}}
\newcommand{\bU}{\ensuremath{{\mathbf U}}}
\newcommand{\bV}{\ensuremath{{\mathbf V}}}
\newcommand{\bW}{\ensuremath{{\mathbf W}}}
\newcommand{\bX}{\ensuremath{{\mathbf X}}}
\newcommand{\bY}{\ensuremath{{\mathbf Y}}}
\newcommand{\bZ}{\ensuremath{{\mathbf Z}}}



%%%%%%%%%%%%Bold roman lower case%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\ba}{\ensuremath{{\mathbf a}}}
\newcommand{\bbe}{\ensuremath{{\mathbf b}}}
\newcommand{\bc}{\ensuremath{{\mathbf c}}}
\newcommand{\bd}{\ensuremath{{\mathbf d}}}
\newcommand{\be}{\ensuremath{{\mathbf e}}}
\newcommand{\beff}{\ensuremath{{\mathbf f}}}
\newcommand{\bg}{\ensuremath{{\mathbf g}}}
\newcommand{\bh}{\ensuremath{{\mathbf h}}}
\newcommand{\bi}{\ensuremath{{\mathbf i}}}
\newcommand{\bj}{\ensuremath{{\mathbf j}}}
\newcommand{\bk}{\ensuremath{{\mathbf k}}}
\newcommand{\bl}{\ensuremath{{\mathbf l}}}
\newcommand{\bem}{\ensuremath{{\mathbf m}}}
\newcommand{\bn}{\ensuremath{{\mathbf n}}}
\newcommand{\boh}{\ensuremath{{\mathbf o}}}
\newcommand{\bp}{\ensuremath{{\mathbf p}}}
\newcommand{\bq}{\ensuremath{{\mathbf q}}}
\newcommand{\br}{\ensuremath{{\mathbf r}}}
\newcommand{\bs}{\ensuremath{{\mathbf s}}}
\newcommand{\bt}{\ensuremath{{\mathbf t}}}
\newcommand{\bu}{\ensuremath{{\mathbf u}}}
\newcommand{\bv}{\ensuremath{{\mathbf v}}}
\newcommand{\bw}{\ensuremath{{\mathbf w}}}
\newcommand{\bx}{\ensuremath{{\mathbf x}}}
\newcommand{\by}{\ensuremath{{\mathbf y}}}
\newcommand{\bz}{\ensuremath{{\mathbf z}}}









%%%%%%%%%%%%%%%%%%%%%Bold numerals%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\bzero}{\ensuremath{{\mathbf 0}}}
\newcommand{\boo}{\ensuremath{{\mathbf 0}}}
\newcommand{\bdo}{\ensuremath{{\mathbf 0}}}
\newcommand{\bone}{\ensuremath{{\mathbf 1}}}
\newcommand{\btwo}{\ensuremath{{\mathbf 2}}}
%%%%%%%%%Calligraphic capitals%%%%%%%%%%%%%%%%%%%%%%


\newcommand{\cA}{\ensuremath{{\mathcal A}}}
\newcommand{\cB}{\ensuremath{{\mathcal B}}}
\newcommand{\cC}{\ensuremath{{\mathcal C}}}
\newcommand{\cD}{\ensuremath{{\mathcal D}}}
\newcommand{\cE}{\ensuremath{{\mathcal E}}}
\newcommand{\cF}{\ensuremath{{\mathcal F}}}
\newcommand{\cG}{\ensuremath{{\mathcal G}}}
\newcommand{\cH}{\ensuremath{{\mathcal H}}}
\newcommand{\cI}{\ensuremath{{\mathcal I}}}
\newcommand{\cJ}{\ensuremath{{\mathcal J}}}
\newcommand{\cK}{\ensuremath{{\mathcal K}}}
\newcommand{\cL}{\ensuremath{{\mathcal L}}}
\newcommand{\cM}{\ensuremath{{\mathcal M}}}
\newcommand{\cN}{\ensuremath{{\mathcal N}}}
\newcommand{\cO}{\ensuremath{{\mathcal O}}}
\newcommand{\cP}{\ensuremath{{\mathcal P}}}
\newcommand{\cQ}{\ensuremath{{\mathcal Q}}}
\newcommand{\cR}{\ensuremath{{\mathcal R}}}
\newcommand{\cS}{\ensuremath{{\mathcal S}}}
\newcommand{\cT}{\ensuremath{{\mathcal T}}}
\newcommand{\cU}{\ensuremath{{\mathcal U}}}
\newcommand{\cV}{\ensuremath{{\mathcal V}}}
\newcommand{\cW}{\ensuremath{{\mathcal W}}}
\newcommand{\cX}{\ensuremath{{\mathcal X}}}
\newcommand{\cY}{\ensuremath{{\mathcal Y}}}
\newcommand{\cZ}{\ensuremath{{\mathcal Z}}}
\newcommand{\cs}{\ensuremath{{\mathcal s}}}


%%%%%%%%%%%%%%%%%%%%Sans Serif%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\newcommand{\sA}{\ensuremath{{\mathsf A}}}
\newcommand{\sB}{\ensuremath{{\mathsf B}}}
\newcommand{\sC}{\ensuremath{{\mathsf C}}}
\newcommand{\sD}{\ensuremath{{\mathsf D}}}
\newcommand{\sE}{\ensuremath{{\mathsf E}}}
\newcommand{\sF}{\ensuremath{{\mathsf F}}}
\newcommand{\sG}{\ensuremath{{\mathsf G}}}
\newcommand{\sH}{\ensuremath{{\mathsf H}}}
\newcommand{\sI}{\ensuremath{{\mathsf I}}}
\newcommand{\sJ}{\ensuremath{{\mathsf J}}}
\newcommand{\sK}{\ensuremath{{\mathsf K}}}
\newcommand{\sL}{\ensuremath{{\mathsf L}}}
\newcommand{\sM}{\ensuremath{{\mathsf M}}}
\newcommand{\sN}{\ensuremath{{\mathsf N}}}
\newcommand{\sO}{\ensuremath{{\mathsf O}}}
\newcommand{\sP}{\ensuremath{{\mathsf P}}}
\newcommand{\sQ}{\ensuremath{{\mathsf Q}}}
\newcommand{\sR}{\ensuremath{{\mathsf R}}}
\newcommand{\sS}{\ensuremath{{\mathsf S}}}
\newcommand{\sT}{\ensuremath{{\mathsf T}}}
\newcommand{\sU}{\ensuremath{{\mathsf U}}}
\newcommand{\sV}{\ensuremath{{\mathsf V}}}
\newcommand{\sW}{\ensuremath{{\mathsf W}}}
\newcommand{\sX}{\ensuremath{{\mathsf X}}}
\newcommand{\sY}{\ensuremath{{\mathsf Y}}}
\newcommand{\sZ}{\ensuremath{{\mathsf Z}}}





%%%%%%%%%%%%%%%%%%%%%%Bold Greek%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\newcommand{\bmu}{\ensuremath{{\boldsymbol \mu}}}
\newcommand{\btheta}{\ensuremath{{\boldsymbol \theta}}}
\newcommand{\bsigma}{\ensuremath{{\boldsymbol \sigma}}}
\newcommand{\btau}{\ensuremath{{\boldsymbol \tau}}}
\newcommand{\bphi}{\ensuremath{{\boldsymbol \phi}}}
\newcommand{\bomega}{\ensuremath{{\boldsymbol \omega}}}
\newcommand{\balpha}{\ensuremath{{\boldsymbol \alpha}}}
\newcommand{\bbeta}{\ensuremath{{\boldsymbol \beta}}}
\newcommand{\bSigma}{\ensuremath{{\boldsymbol \Sigma}}}
\newcommand{\bzeta}{\ensuremath{{\boldsymbol \zeta}}}
\newcommand{\beps}{\ensuremath{{\pmb \varepsilon}}}
\newcommand{\bdelta}{\ensuremath{{\pmb \delta}}}
\newcommand{\bnu}{\ensuremath{{\boldsymbol \nu}}}
\newcommand{\bxi}{\ensuremath{{\boldsymbol \xi}}}
\newcommand{\boldeta}{\ensuremath{{\boldsymbol \eta}}}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Hats on Romans%%%%%%%%%%%%%%%%%
\newcommand{\bxdot}{\ensuremath{\dot{\bx}}}
\newcommand{\db}[1]{\bar{\bar{#1}}}
\newcommand{\hB}{\ensuremath{{\hat{B}}}}
\newcommand{\hk}{\ensuremath{{\hat{k}}}}
\newcommand{\hA}{\ensuremath{{\hat{A}}}}
\newcommand{\Ahat}{\ensuremath{{\widehat{A}}}}
\newcommand{\Phat}{\ensuremath{{\widehat{P}}}}
\newcommand{\hsigmasq}{\ensuremath{{\widehat{\sigma^2}}}}
\newcommand{\jhat}{\ensuremath{\widehat{j}}}
\newcommand{\bxhat}{\ensuremath{\widehat{\bx}}}
\newcommand{\bphat}{\ensuremath{\widehat{\bp}}}
\newcommand{\bptilde}{\ensuremath{\widetilde{\bp}}}
\newcommand{\ptilde}{\ensuremath{\widetilde{p}}}
\newcommand{\bmtilde}{\ensuremath{\widetilde{\bm}}}
\newcommand{\bCtilde}{\ensuremath{\widetilde{\bC}}}
\newcommand{\bytilde}{\ensuremath{\widetilde{\by}}}
\newcommand{\cop}{\ensuremath{\text{cp}}}
\newcommand{\coq}{\ensuremath{\text{cq}}}
\newcommand{\bPhat}{\ensuremath{\widehat{\bP}}}

%%%%%%%%%%%%%%%%%%%%%%Hats on greeks%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand\halpha{\hat{\alpha}}
\newcommand{\htheta}{\ensuremath{{\hat{\theta}}}}
\newcommand{\hbtheta}{\ensuremath{{\hat{\btheta}}}}
\newcommand{\chtheta}{\ensuremath{{\check{\theta}}}}
\newcommand{\htau}{\ensuremath{{\hat{\tau}}}}
\newcommand{\muhat}{\ensuremath{{\hat{\mu}}}}
\newcommand{\hSigma}{\ensuremath{{\hat{\Sigma}}}}
\newcommand{\pitilde}{\ensuremath{\widetilde{\pi}}}

%%%%%%%%%%%%%%%Vectors%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vect}[2]{(#1_1,\; #1_2,\;\ldots\;,#1_{#2})}
\newcommand{\vecto}[2]{(#1_0,\; #1_2,\;\ldots\;,#1_{{#2}-1})}
\newcommand{\uvect}[2]{#1_1,\; #1_2,\;\ldots\;,#1_{#2}}
\newcounter{ccnt} %for vectors
\newcommand{\nuvect}[2]{\setcounter{ccnt}{1}%
                         \ensuremath{\whiledo{\value{ccnt}<#2}{%
                         #1_{\theccnt},\,\stepcounter{ccnt}}#1_{#2}}\xspace}
\newcommand{\nuvecto}[2]{\setcounter{ccnt}{0}%
                         \ensuremath{\whiledo{\value{ccnt}<#2}{%
                         #1_{\theccnt},\,\stepcounter{ccnt}}#1_{{#2}}}\xspace}



%%%%%%%%%%%%%%%%%%%Theorems%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{thm}{Theorem}
\newtheorem{corrol}{Corollary}
\newtheorem{lem}{Lemma}
\newtheorem{defn}{Definition}
%\newtheorem{problem}{Problem}
\newtheorem{remark}{Remark}


%%%%%%%%%%%%%%%%%%%%Figures%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\myfig}[4][h!]{
\begin{figure}[#1]
  \begin{center}
\resizebox{5.5in}{5.5in}{
\includegraphics{#2}}
    \caption{#3}
    \label{#4}
  \end{center}
\end{figure}
}

%%%%%%%%%%%%%%%%%%%%Environments%%%%%%%%%%%%%%%%%%%%%%%%%

\renewenvironment{cases}{\left\{\begin{array}{ll}}{\end{array}\right.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newenvironment{tightlist}
  { \begin{itemize}
    \vspace{-4pt}
    \setlength{\topsep}{0pt}
    \setlength{\partopsep}{0pt}
    \setlength{\parsep}{0pt}
    \setlength{\itemsep}{0pt}
    \setlength{\parskip}{0pt}
  }
  { \vspace{-4pt}
    \end{itemize}
  }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%Background Picture%%%%%%%%%%%%%%%%%%%%
%\usepackage{eso-pic,graphicx}
\newcommand\BackgroundPicture[2]{%
\setlength{\unitlength}{1pt}%
default \put(0,\strip@pt\paperheight){%
\parbox[t][\paperheight]{\paperwidth}{%
\vfill
\centering\includegraphics[angle=#2]{#1}
\vfill
}}} %
%\AddToShipoutPicture{\BackgroundPicture{rose}{45}}% here or anywhere
%\AddToShipoutPicture{\BackgroundPicture{rose}{55}}
%in LateX preamble or anywhere in the text an image is printed on the actual
%page where this command is located and all following one. 
%The arguments
%1. the first one is the name of the image with relative path
%2. the second is the angle for the rotating of the image




























